AMath350.W14.A3.Sol

# 4 4 dt et 2 sin2 4 integratingdt factor e t 21 sin2

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Unformatted text preview: sin(2 t) . ( (a) The a commodity eD 1+cos(2 t = e demand t)20+4p)) by 5.6.(Consider DE is Factor: (withS ) = dt)(20+8p+ 1p3/2 curves given= (4p p3/2 ), with IC p(0) = p0 . 4 4 dt = et 2 sin(2 . 4 Integratingdt Factor: e t+ 21 sin(2 t) dp t 2p Equilibrium solutions:(1 p) cos(2⇡20 ++41 sp(4 t) p p1/2 ) 5(1 + = 0, 16. et+ 21 sin(2 t) 4+ p ) p cos(2⇡ t)) S ( p =3/2 t= e0 ) in(2 e + )) = 1 1 0 3 0 dt Bernoulli+equationdp D(pn +=3/20⇡Let tv 2=3in(2 3t/2 == 1/2 ) cos(2⇡ t))pt+ /12 pin(2Using the 2. t 8e 1 p1 et 2 sin(2 t) with(1 =cos(2 +))p + p s/2 ) p p 5(1 + v = 1 2 e 2 s . t) +) d t+ 1 sin(2 t) dt DE: e2 p = 5(1 + cos(2⇡ t))et+ 2 sin(2 t) dt d rate of sin(2 t) 1 The rate of increase of price is 1/4 the t+e1t1sin(2 decrease = 1quantity and t))et+ 21 sin(2 t) +2 1 of 5(1 +) cos(2⇡ the t) p t+ sin(2 t + C v 0 = e 2 p 1/2p+ = 5e 2 dt initial price is p0 . ) 1 2 in(2 General Solution:21psin(2 = p 5 = C 5et +221 ssin(2 t)t) + C t8 et et+ (1) t) 1 a) (Set up an initial value problem = that models thisLinear DE and ﬁnd all situation 1 v+ IC: p(0) = p ) p0 = 5 C )General Solution: p(t) = 5 C e t 2 sin(2 t) C = 5 p0 . equilibrium 0 solutions of the diﬀerential2in(2 t) 8 equation. Then ﬁnd the solution 1 Solution of IVP: p(t) = 5 (5 p )e t 2 s . 1 of the(0) =form: v 00+ lim = t1 .= 50C = 5 p0 . solutions:2 dt(t) =t/2 . IC: p initial 0value problem. ) Integrating factor: e 1/ p = e 5. Standard behaviour: =25 p(C ) . Equilibrium v8 Longterm p ) p ) t Solution: IVP: p(t) = 5 (5 p0 )e t 21 sin(2 t) . Solution of 1 t/2 /2 1 t/2 dp 1 t/2 1 0 = (20+4p)) = 1 (4p p / t = 5. e 6. (a) Longterm behaviour: Slimvp(t) = 5.v p3Equilibrium solutions: 3p2(),)with IC p(0) = p0 . The DE is = (D e) = + e p (20+8 t 8 dt 4 42 4 Equilibriumdp solutions: 4p p3/21= t/2 p(4 p1/2 )2) p = 0, 16. 1 d 0) 1 t/ 1 Bernoulli equation with n ) =dt [e Let ]vp= p1 82e 2 (20+42 )) =0 = p1 p p3/22p0 . with IC p(0) = p0 . 6. (a) The DE is = (D S = 3/2. v = 3/ 3/ = p 1/ p ) v (4 2 3/ ), Using the (20+8 dt 4 4 4 DE: Equilibrium solutions: 4p p3/2et/2 v ) p(4 et/2 1/2 C) p = 0, 16. = 0 = 1 p+ ) 1 0 4 Bernoulli equation with n v= 3/2. 1 p 1/2 + 1p1 3/2 = p...
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